Discrete random uniform distribution

Standard series results

It’s important to remember our standard series results when working with discrete random uniform distributions. The main two are these ones:

• \sum_{r=1}^n r = \frac12n(n+1)
• \sum_{r=1}^n r^2 = \frac16n(n+1)(2n+1)

Transformations

Another reminder of how we transform discrete random probabilities.

If Y=aX+b is a transformation of the DRV X:

• E(Y)=aE(X)+b
• Var(Y)=a^2Var(X)

Rectangular distribution

If we have something that looks like a uniform distribution, but it’s a rectangle instead of vertical lines of the same height, then it’s not a uniform distribution, but instead a rectangular distribution.

You should be able to tell: the graphs of rectangular distributions are, well, rectangles.

Notation for distributions

We can represent a probability distribution using a notation like this:

• X \sim A(b, c, ...)

That means that X is an A distribution (whatever A may be) and we need to know the parameters: b, c, etc. to be able to work with it.

Uniform distribution notation

X \sim U(n)

This means that X is a discrete random variable which follows a uniform distribution, from 1 to n.

Example: rolling a die

For a normal, fair, 6-sided die, our distribution is:

X \sim U(6)

Let’s say we were to write a table of our probabilities for this distribution. It would look like this:

Let’s calculate our expected value and variance for the distribution:

• E(X)=\sum xp = \frac{21}{6} = 3.5
• E(X^2)=\sum x^2p = \frac{91}{6}
• E(X)^2 = \left(\frac{21}{6}\right)^2
• Var(X)=E(X^2)-E(X)^2 = \frac{91}{6} - \left(\frac{21}{6}\right)^2 = \frac{35}{12}

General formula

We’ve calculated the expected value and variance for a specific example, but we can do it more generally. Let’s say that P(X=r)=\frac1n for r=1, 2, ..., n.

We can put this into a table to visualise it:

To calculate our expected value:

• E(X)=\sum xp=\frac1n+\frac2n+\frac3n+...+\frac nn
• E(X)=\frac{\sum_{r=1}^n r}n
• E(X)=\frac{n(n+1)}{2n}
• E(X)=\frac{n+1}{2}
• So the expected value of a discrete random variable which follows a uniform distribution from 1 to n is \frac{n+1}{2}.

Now, our variance. This one’s a little more complicated:

• E(X^2)=\sum x^2p
• E(X)^2 = \left(\sum xp\right)^2
• Var(X)=E(X^2)-E(X)^2
• Var(X)=\sum x^2p-(\sum xp)^2
• Var(X)=\frac{\frac16n(n+1)(2n+1)}n-\frac14(n+1)^2
• Var(X)=\frac{(n+1)(2n+1)}{6}-\frac{(n+1)^2}{4}
• Var(X)=\frac1{12}(n+1)(2(2n+1)-3(n+1))
• Var(X)=\frac1{12}(n+1)(4n+2-3n-3)
• Var(x)=\frac1{12}(n+1)(n-1)
• Var(X)=\frac1{12}(n^2-1)

For a discrete random variable X which follows a uniform distribution from 1 to n:

• E(X)=\frac{n+1}{2}
• Var(X)=\frac1{12}(n^2-1)

Probability of a specific value

To find the probability of getting x from a discrete random variable following a uniform distribution:

P(X=x)=\frac1n\quad\text{for values in set} 0\quad\quad\quad\quad\quad\quad\quad \text{otherwise}

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